To represent an arbitrary signal without error, it is well known that a signal must be measured at a rate that is at least twice the highest frequency (Nyquist rate). However, it is known certain signals can be compressed after measuring. It has been shown that measuring such signals and then compressing the signals wastes resources. Instead, compressive sensing can be used to efficiently acquire and reconstruct signals that are sparse or compressible. Compressive sensing leverages the structure of the signals to enable measuring at rates significantly lower than the Nyquist rate. Compressive sensing can use randomized, linear, non-adaptive measurements, followed by non-linear reconstruction using convex optimization, specifically, l1 norm minimization, or using greedy searches.
Digital measurements of signals are quantized to a finite number of bits. e.g., only the most significant (sign) bit. However, reconstruction a signal from one bit measurements is difficult. One method combines the principle of consistent reconstruction with l1 norm minimization on a unit sphere to reconstruct the signal.